NS8-1 Factors and Multiples

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1 NS- Factors and Multiples The multiples of a number are the numbers you say when counting by that number. is a multiple of both and 0 is a multiple of both 0 and = 0 = 0 and are both factors of 0 and are both factors of 0. List the first few multiples of these numbers. : 0,,,,,, b) :,,,,,, c) :,,,,,,. Look at the lists you made in Question. Is a multiple of? How do you know? b) Is a multiple of? How do you know? c) Is 0 a multiple of? Of? Of?. Write 0 as a multiple of : 0 = b) Which whole numbers is 0 a multiple of? Explain. COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED. Rewrite each statement in a way that means the same thing but uses the word factor. 0 is a multiple of. b) is a multiple of. c) 0 is a multiple of. d) is a multiple of. e) is not a multiple of. f) Every number is a multiple of. g) Every number is a multiple of itself. h) 0 is a multiple of any number.. Rewrite each statement in a way that means the same thing but uses the word multiple. is a factor of. b) is a factor of. c) is a factor of 0. d) Every number is a factor of 0. e) is a factor of. f) is a factor of every number. g) is a factor of. h) Any number is a factor of itself. Number Sense -

2 . Alana wants to find all pairs of numbers that multiply to give 0. She lists each number from to 0 in a chart. She looks for a second number that multiplies with the first to give 0. Why didn t Alana list any number greater than 0 in the first column of her table? b) Why didn t Alana list 0 in the first column of her table?. Use Alana s method to find all pairs of numbers that multiply to give the number in bold. st nd b) st nd c) st nd st nd 0 0. Cross out the pairs that are repeated in Question.. Connor makes a chart to list all the factors of 0. He doesn t want to write and check all the numbers from to 0. He starts his list as follows: Connor knows that = 0. He thinks that if = 0, then must be less than. Explain his thinking. b) Explain why Connor s list is complete. st nd Connor used this chart to help him identify pairs that multiply to. Why did he know that his search was complete as soon as he found a pair with both numbers the same?. Find all pairs of numbers that multiply to 0. st nd COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED Number Sense -

3 NS- LCMs and GCFs. Mark the multiples of each number on the number lines. : : : : is a multiple of every number. Not counting 0, find the first common multiples of: and, b) and, c) and, d) and, The lowest common multiple (LCM) of two or more numbers is the smallest number (not 0) that is a multiple of the numbers.. Look at your answers to Question. What is the LCM of: and b) and c) and d) and. Find the lowest common multiple of each pair of numbers. and b) and 0 c) and d) and :,,,,, : : : :, 0,, 0 0: : : LCM = LCM = LCM = LCM = e) and 0 f) and g) and h) and i) and j) and k) and 0 l) and 0 m) and n) and COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED. How can you find the second common multiple of two numbers from the first? b) The first common multiple of and is. What is the second common multiple?. Find all the factors of each number by dividing the number by the whole numbers in increasing order divide by,,,,, and so on. How do you know when you can stop dividing? b) c) d) e) 0,,, Number Sense -

4 The greatest number that is a factor of two or more numbers is called the greatest common factor (GCF) of the numbers.. Use your answers to Question. Find the greatest common factor of: and b) and c) and 0 d) and e) and f) and g), and h), and 0 Two numbers are called consecutive if one number is the next number after the other. Example: and are consecutive because is the next number after. INVESTIGATION What is the GCF of two consecutive numbers? A. Find the factors of each number and then the GCF of each pair. and b) and c) and d) and :,,, : : : :,,, : : : GCF: GCF: GCF: GCF: B. Make a conjecture about the GCF of any two consecutive numbers. C. Test your conjecture on two consecutive numbers of your choice: and GCF: and are multiples of. So + and are multiples of, too! = = + = =. Rewrite the conclusion in the box using the word factor instead of multiple: is a factor of both and, so is a factor of both and b) Draw pictures to show that any factor of both and 0 is also a factor of both 0 + and 0. c) Explain why any common factor of and 00 must divide the sum d) Explain why any common factor of and 00 must divide the difference 00. e) Without finding the factors of and 00, explain why their GCF is. COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED Number Sense -

5 INVESTIGATION How are the GCF, the LCM, and the product of two numbers related? A. Complete the chart. Include three more values of your choice for a and b. a b a b GCF LCM GCF LCM 0 0 B. Which two columns are the same in every row? and C. Write an expression for the LCM in terms of a b and GCF. LCM = D. When the LCM is the same as the product, what is the GCF? E. Choose two more pairs of numbers a and b where a is a factor of b, and complete the chart. a b a b GCF LCM GCF LCM COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED Which columns are equal? GCF =, LCM = and GCF LCM = Number Sense -

6 NS- Prime Numbers A prime number has exactly two distinct factors: itself and. A composite number has more than two distinct factors: itself, at least one number other than itself, and.. How many distinct factors does the number have? Is a prime number?. List all prime numbers less than 0:. List all composite numbers between 0 and 0:. What is the greatest prime number less than 0?. Circle the prime numbers in each list. b) 0 c) 0 d). List all the factors of each number. :,, b) : c) : d) : e) : f) : g) 0: h) : i) 0: j) :. Put a check mark in front of the numbers that are composite numbers. 0. Write a number between 0 and 0 that has two factors b) four factors c) five factors. The prime numbers and differ by. Find three other pairs of prime numbers less than 0 that differ by : 0. Write three consecutive numbers which are also all composite numbers: COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED Number Sense -

7 . Eratosthenes was a Greek scholar who was born over 000 years ago in what is now Libya. He developed a method to systematically identify prime numbers. It is called Eratosthenes Sieve. Follow these directions to use Eratosthenes Sieve: Shade the number (it is not prime). b) Circle,,, and all the primes less than 0. c) Shade all the remaining multiples of. d) Shade all the remaining multiples of. e) Shade all the remaining multiples of. f) Shade all the remaining multiples of. g) Circle the next uncircled number (). Note that all multiples of less than 00 (,,, ) are already shaded because they have a factor less than 0. h) Circle the next uncircled number. How do you know all multiples of that number less than 00 are already shaded? i) Now circle all the remaining numbers. You ve just used Eratosthenes Sieve to circle all the prime numbers from to 00!. How many prime numbers are there between 0 and 0? COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED. Solve these riddles. I am a prime number less than 00. If you add 0 or 0 to me, the result is prime. What number am I? b) I am a prime number less than 00. My digits add to. What number am I? c) I am a prime number less than 00. My tens digit is one more than my ones digit. What number am I? d) I am a prime number between 0 and 0. If you reverse my digits, the result is a larger prime number. What number am I? Number Sense -

8 NS- Prime Factorizations Any composite number can be written as a product of prime numbers. This product is called the prime factorization of the original number. 0 is not a prime factorization of 0 because the number 0 is composite is a prime factorization of 0 You can find a prime factorization for a number by using a factor tree. Here is how you can make a factor tree for the number 0: Step : Find any pair of numbers (not including ) that multiply to give 0. 0 Step : Repeat Step for the numbers on the branches of the tree is a prime number, so you can leave it as is.. Complete the factor trees. b) c). Write a prime factorization for each number. 0 = 0 = b) = c) = d) =. Use a factor tree to find a prime factorization for each number. 0 b) c) d) e). Here are some branching patterns for factor trees: Can you find a factor tree for the number that has a different branching pattern from the tree in Question c)? COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED Number Sense -

9 NS- Prime Factorizations and GCFs INVESTIGATION How does the prime factorization of a number compare to the prime factorization of its factors? A. Write the prime factorization of and all its factors: Factors of Prime Factorization B. How many s are in the prime factorization of? Does any factor of have more s in its prime factorization than does? C. How many s are in the prime factorization of? Does any factor of have more s in its prime factorization than does? D. Finish the sentences below by writing at least or at most. Any factor of must have as many s in its prime factorization as does. Any factor of must have as many s in its prime factorization as does. E. Does have a in its prime factorization? Does any factor of have a in its prime factorization? Explain why this is so. COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED. The prime factorization of 0 is. Without doing any calculations, circle the products that show factors of 0: How did you decide which products to circle? Number Sense -

10 . Find the prime factorizations of and. Do the rough work in your notebook. = = b) Any factor of must have in its prime factorization at most: two s, one s, one s c) Any factor of must have in its prime factorization at most: s, s d) Can a common factor of and have any s in its prime factorization? How do you know? e) Any common factor of and must have in its prime factorization at most: s, s f) The prime factorization of the greatest common factor (GCF) of and is: So the GCF of and is.. The prime factorization of each number is given. Match up as many pairs of common prime factors as you can. Then find the prime factorization of the GCF and calculate the GCF. = b) 0 = = 0 = GCF = = GCF = = c) = d) 0 = = 0 = GCF = = GCF = =. Write a prime factorization for each number, then find the GCF of each pair. and b) and 0 c) and 0 d) and e) 0 and. Find the GCF of the numbers., 0, b), 0, 00 c),, 0 COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED 0 Number Sense -

11 NS- Prime Factorizations and LCMs INVESTIGATION How does the prime factorization of a number compare to the prime factorization of its multiples? A. Write the prime factorizations of the first ten multiples of 0 (don t include zero). Multiples of 0 Prime factorizations B. How many s are in the prime factorization of 0? Does any multiple of 0 have fewer s in its prime factorization than 0 does? C. How many s are in the prime factorization of 0? Does any multiple of 0 have fewer s in its prime factorization than 0 does? D. How many s are in the prime factorization of 0? Does any multiple of 0 have fewer s in its prime factorization than 0 does? E. Finish the sentences below by writing at least or at most. Any multiple of 0 must have as many s in its prime factorization as 0 does. Any multiple of 0 must have as many s in its prime factorization as 0 does. Any multiple of 0 must have as many s in its prime factorization as 0 does. F. Does 0 have a in its prime factorization? Does any multiple of 0 have a in its prime factorization? COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED. The prime factorization of 0 is. Without doing any calculations, circle the products that show multiples of 0: How did you decide which products to circle? Number Sense -

12 . Find the prime factorizations of 0 and. Do the rough work in your notebook. 0 = = b) Any multiple of 0 must have in its prime factorization at least: one s, two s, one s c) Any multiple of must have in its prime factorization at least: s, s, s d) Any common multiple of 0 and must have in its prime factorization at least: s, s, s, and s. e) The lowest common multiple (LCM) of 0 and must be: =. Find the prime factorizations of 00 and. 00 = = b) Any multiple of 00 must have in its prime factorization at least: s, and s c) Any multiple of must have in its prime factorization at least: s, s, and s d) Any common multiple of 00 and must have in its prime factorization at least: s, s, s, and s. e) The lowest common multiple (LCM) of 00 and must be: =. The prime factorization of each number is given. Find the prime factorization of the LCM. Then calculate the LCM. 0 = and 0 = So LCM = = b) 0 = and 0 = So LCM = =. Find the prime factorizations of each number. Then find the prime factorization of the LCM and calculate the LCM. and b) and c) 0 and 0 d) and e) and COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED Number Sense -

13 NS- Order of Operations Addition and subtraction are done from left to right. If there are brackets, do the operations in brackets first. Example: + = + = but ( + ) = =. Calculate each expression using the correct order of operations. ( + ) + ( ) + ( ) ( + ) + ( ) ( + ) ( ) b) How many different answers did you get in part?. Add brackets in different ways to get as many different answers as you can. i) ii) + iii) b) How many different answers did you get in part? i) ii) iii) c) Check all that apply. The order of operations affects the answer when the expression consists of addition only subtraction only addition and subtraction Multiplication and division are done from left to right. If there are brackets, do the operations in brackets first. Example: = = but ( ) = =. Evaluate each expression. b) c) 0 ( ) d) ( ). Add brackets in different ways to get as many different answers as you can. i) ii) iii) 0 b) Which expressions in part give the same answer, no matter where you place the brackets? COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED. Do the operation in brackets first. + ( ) b) ( + ) c) ( + ) d) + ( ) = + = e) ( ) f) ( ) g) ( ) h) ( ). Check all that apply. The order of operations affects the answer when the expression combines addition and multiplication subtraction and multiplication addition and subtraction addition and division subtraction and division multiplication and division Number Sense -

14 Mathematicians have ordered the operations to avoid writing brackets all the time. The order is:. Operations in brackets.. Multiplication and division, from left to right.. Addition and subtraction, from left to right Example: + = ( ) + ( ) but ( + ) = + = = = Notice that the brackets in the first expression are not necessary.. Evaluate each expression. Use the correct order of operations. b) + c) d) 0 + e) + f) g) ( ) h) ( ). Translate the instructions into mathematical expressions. Add and. Then subtract. Then multiply by. ( + ) b) Subtract from. Then multiply by. Then add. c) Multiply and. Then subtract from. Then add.. Write the expressions in words. ( + ) Add and. Then multiply by. b) ( ) Multiply and. Then subtract from. Then c) ( + ) d) ( + ) 0. Calculate the expression in the box in your notebook. Which expression without brackets gives the same answer? ( + ) = or + b) ( ) = or + c) + ( ) = + or + + d) + ( + ) = + + or +. Add brackets in different ways to get as many different answers as you can. i) + ii) + iii) + b) How many different answers did you get in part? i) ii) iii). Rewrite each expression without brackets by changing only operations symbols. Keep the answer the same. ( ) b) ( ) c) ( ) COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED Number Sense -

15 INVESTIGATION What types of expressions can be written without brackets? A. Write the dimensions of the two smaller rectangles in the blanks. Write the area of the large rectangle in two ways: one with brackets and one without. Then write an equation. = + Area = ( + ) and Area = + So ( + ) = + B. Write these expressions without brackets. Draw a picture if it helps. (0 + ) = + b) (0 + ) = + C. Use your answers in B. to find and. D. Write the dimensions of the four smaller rectangles in the blanks. Write the area of the large rectangle in two ways, one with brackets and one without. Then write an equation. = COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED Area = ( + ) ( + ) and Area = So ( + ) ( + ) =. Number Sense -

16 E. Write these expressions without brackets. Draw pictures in your notebook to show your answers. (0 + ) (0 + ) = b) (0 + ) (0 + ) = c) ( + + ) = + + d) ( + + ) = + + F. Use your answers to part E to find and. G. Calculate both sides to determine which equations are true. ( + ) = + ( + ) = = + = + = Is the equation true? b) ( + ) = + ( + ) = = + = + = Is the equation true? c) ( ) = ( ) = = = = Is the equation true? d) ( ) ( ) = ( ) ( ) = = = = Is the equation true? e) 0 ( ) = ( ) = = 0 0 = = Is the equation true? f) ( ) = ( ) = = = = Is the equation true? H. Match each expression with its description. ( + ) Can be written without brackets and does not require writing more numbers. ( + ) Can be written without brackets but requires writing more numbers. ( + ) Cannot be written without brackets (using only +,,, ). COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED Number Sense -

17 NS- Fractions Fractions name equal parts of a whole. This pie is cut into equal parts, and of the parts are shaded. So of the pie is shaded. The numerator tells you how many parts are counted. The denominator tells you how many equal parts are in a whole.. Name the following fractions. b) c). Draw lines to divide each figure into equal parts. Then say what fraction of each figure is shaded. b) c) d). Divide each line into the given parts, as done in part. Thirds b) Halves c) Thirds d) Quarters e) Fifths Fractions can name parts of a set. In this set, of the figures are pentagons, and are circles: are squares,. Fill in the blanks for this set: 0 of the figures are. b) 0 of the figures are. c) of the figures are squares. d) of the figures are triangles. COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED. Fill in the chart. Whole Numbers Whole Numbers from to from 0 to b) What fraction of the whole numbers from to are composite? Prime Numbers Composite Numbers,, c) What fraction of the whole numbers from to are prime? Number Sense -

18 NS- Mixed Numbers Mattias and his friends ate the amount of pie shown. They ate two and three quarter pies altogether (or pies). is called a mixed number because it is a mixture of a whole number and a fraction. whole pies and of another pie. Follow the example to find the mixed number for each picture. b) c) whole pies and whole pie and whole pies and of another pie = pies of another pie = pies of another pie = pies. Write each fraction as a mixed number. b) c) d). Shade the area given by the mixed number. Note: There may be more figures than you need. b) c) d). Sketch: pies b) squares c) pies d) rectangles e) circles. Order from smallest to largest:,,.. Which is closer to : or? Explain. COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED Number Sense -

19 NS-0 Improper Fractions Huan-Yue and her friends ate quarter-sized pieces of pizza: = Altogether, they ate pizzas. improper fraction mixed number When the numerator of a fraction is larger than the denominator, the fraction represents more than a whole. Such fractions are called improper fractions.. Write these fractions as improper fractions. b) c) d) e) f). Shade one piece at a time until you have shaded the amount given by the improper fraction. b) c) d) COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED. Sketch: pies b) squares c) parallelograms d) rectangles e) circles. Order from smallest to largest:,,, 0.. Which fractions are improper fractions? How do you know? b) c) Number Sense -0 d)

20 NS- Mixed and Improper Fractions. Write these fractions as mixed numbers and as improper fractions. b) c) d) e) f). Shade the amount of pie given by the mixed fraction. Then write an improper fraction for the amount. b) Improper fraction: Improper fraction:. Shade the amount of area given by the improper fraction. Then write a mixed number for the amount. b) Mixed number: Mixed number: c) d) Mixed number:. Draw a picture to find out which fraction is greater.,, b),, c) Mixed number:,,. How could you use division to find out how many whole pies are in Explain. d) of a pie?,, COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED 0 Number Sense -

21 NS- More Mixed Numbers There are quarter pieces There are ( ) quarters There are ( ) quarters in pie. in pies. in pies. pieces ( ) + extra piece How many quarter pieces are in pies? So there are quarter pieces altogether.. Find the number of halves in each amount. pie = halves b) pies = halves c) pies = halves d) pies = halves e) pies = halves f) pies =. Find the number of thirds in each amount. pie = thirds b) pies = thirds c) pies = thirds d) pies = thirds e) pies = f) pies =. A box holds cans, so each can is a fourth. Find the number of cans in each amount. boxes hold cans b) boxes hold cans c) boxes hold cans. If a bag holds peas, then bags hold peas b) bags hold peas c). Write the mixed numbers as improper fractions. bags hold peas = b) = c) = d) = e) =. Envelopes come in packs of. Alice used packs. How many envelopes did she use? COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED. Maia and her friends ate BONUS. How many quarters are there in dollars?. Cindy needs cups of flour. How many scoops of cup A would she need? b) How many scoops of cup B would she need? pizzas. How many quarter-sized pieces did they eat? A cup B cup Number Sense -

22 NS- More Mixed Numbers and Improper Fractions How many whole pies are there in pies? There are pieces altogether, and each pie has pieces. So you can find the number of whole pies by dividing by : = remainder There are whole pies and quarter left over: =. Find the number of whole pies in each amount by dividing. pies = whole pies b) pies = whole pies c) pies = whole pies d) pies = whole pies e) pies = whole pies f) 0. Find the number of whole pies and the number of pieces remaining by dividing. b) c) d) pies = whole pies and half pie = pies pies = whole pies and half pie = pies pies = whole pies and third = pies pies = whole pies and fourth = pies pies = whole pies. Write the following improper fractions as mixed numbers. b) c) d) e) f) g) 0 h). Write a mixed number and improper fraction for the total number of litres: L. Write a mixed number and improper fraction for the length of the rope: m. Order from smallest to largest:,,.. Between which two whole numbers is?. How much greater than a whole is each fraction?. Which fractions are greater than but less than? b) c) d) b) c) d) e) 0 COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED Number Sense -

23 NS- Comparing Fractions Introduction. Shade the given amount in each pie. Then circle the greater fraction in each pair. b) c) 0 0. Two fractions have the same denominators (bottoms) but different numerators (tops)? How can you tell which fraction is greater?. Shade the given amount in each pie. Then circle the greater fraction in each pair. b) 0 c) 0. Two fractions have the same numerators (tops) but different denominators (bottoms). How can you tell which fraction is greater?. Write the fractions in order from least to greatest.,, b),,, c),, d) 0, 0, 0, 0 e),, f),, COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED BONUS. Which fraction is greater? How do you know? or b) or Number Sense -

24 NS- Equivalent Fractions. Compare the fractions by shading to see which is more. Write > (more than), < (less than), or = (equal). b) c) > d) e) f) 0 0 Two fractions are said to be equivalent if they represent the same amount.. List two pairs of equivalent fractions from Question. = and =. Group the squares to make an equivalent fraction. How many of the equal larger groups are shaded? b) c) = 0 = 0 =. Write three equivalent fractions for the amount shaded here:. Draw lines to cut the pies into: equal pieces equal pieces equal pieces b) Then fill in the numerators of the equivalent fractions: = = =. Make an equivalent fraction by cutting each piece into the same number of parts. = b) = c) = d) = COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED Number Sense -

25 NS- Comparing Fractions Using Equivalent Fractions. Write six equivalent fractions by skip counting to find the numerators. = = = = = = b) = 0 = = 0 = = 0 =. Find two fractions with the same denominators from the lists in Question. and Which fraction is greater: or? How do you know? When you multiply the numerator and denominator of a fraction by the same number, you create an equivalent fraction. = = 0 You are cutting each piece into parts. Create an equivalent fraction with denominator by multiplying the numerator and denominator by the same number: = b) = c) = d) = e) = f) = g) = h) =. Write the fractions from Question in order from smallest to largest.. Write several fractions equivalent to. = = = = 0 = = = = = 0 COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED b) How much more than a half is each fraction below? is more than 0 is more than is more than is more than c) Write all the given fractions from part b) in order from smallest to largest. is more than is more than Number Sense -

26 NS- Adding and Subtracting Fractions Introduction. Imagine moving the shaded pieces from pies A and B into pie plate C. Show how much of pie C would be filled, then write a fraction for pie C. A B C + =. Imagine pouring the liquid from cups A and B into cup C. Shade the amount of liquid that would be in C. Then complete the addition statements. L A L B L C L A L B L C + = + =. Add. + = b) + = c) + = d) + = e) + = f) 0 + = g) + = h) + =. Show how much pie would be left if you took away the amount shown. Then complete the fraction statement. b) = =. Subtract. = b) e) 0 = f) = c) = g) = d) = h) = =. Calculate. + + = b) + = c) 0 + = COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED Number Sense -

27 NS- Adding and Subtracting Fractions To add fractions with different denominators: Step : Find the LCM of the denominators. + Multiples of : 0,,,,,, Multiples of : 0,, 0,, 0,, 0 LCM (, ) = Step : Create equivalent fractions with that denominator. + = The LCM of the denominators is called the lowest common denominator (LCD) of the fractions. = + = +. Find the LCD of each pair of fractions. Then show what numbers you would multiply the numerator and denominator of each fraction by in order to add. LCD = b) LCD = c) LCD = d) LCD = e) LCD = f) LCD = g) LCD = h) LCD = Add or subtract the fractions by changing them to equivalent fractions with denominator equal to the LCD of the fractions. + b) + c) d) = = = = = = = = COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED e) + f). Add or subtract. f) + b) g) c) g) + + h) d) h) + + i) + e) i) + 0 Number Sense -

28 A fraction is reduced to lowest terms when the greatest common factor of its numerator and denominator is the number. is not in lowest terms because the GCF of and is. Factors of :,,, Factors of :,,, is in lowest terms because the GCF of and is. Factors of :, Factors of :,,. Find the GCF of the numerator and denominator. Is the fraction in lowest terms? Write yes or no. b) c) GCF = GCF = GCF = GCF = GCF = no d) 0 e) 0 f) 0 g) h) i) j) To reduce a fraction to lowest terms: Step : Find the GCF of the numerator and denominator Step : Divide both the numerator and denominator by the GCF.. Reduce the fractions below by dividing the numerator and the denominator by their GCF. 0 = b) = c) = d) = e) = f) = g) = h) 0 =. Add or subtract, then reduce your answer to lowest terms. e) = + 0 = = 0 + b) f) c) d) 0 g) + h) + COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED Number Sense -

29 NS- Adding and Subtracting Mixed Numbers. Add or subtract. + = b) = c) + = d) + = e) = f) =. Add or subtract by changing the fractions to equivalent fractions. + b) c) = = + = + + = + = = d) + e) f) +. + =. How can you simplify this answer?. is greater than. How can you subtract?. Change the improper fractions to mixed numbers. i) = ii) = iii) = iv) = COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED b) Rewrite each mixed number to make the improper fraction a proper fraction. i) = + ii) = iii) = = + = = = = = Number Sense -

30 c) Add by changing the fractions to equivalent fractions. Simplify your answer as in part b). i) + ii) + iii) + = + + = + + = =. Rewrite each mixed number by regrouping whole as a fraction. Example: = + = i) ii) iii) iv) v) vi) b) Subtract by rewriting the first mixed number as in part : i) = 0 0 = 0 0 = 0 ii). Add or subtract by first changing the mixed numbers to improper fractions. + b) c) + d) 0 = + 0 = + = 0 =. Sonjay cycled km in the first hour, km the second hour, and km the third hour. How many kilometres did he cycle in the three hours?. A cafeteria sold cheese pizzas, vegetable pizzas, and deluxe pizzas at lunchtime. How many pizzas did they sell altogether? 0. Gerome bought metres of cloth. He used to make a banner. How many metres of cloth were left over? COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED 0 Number Sense -

31 NS-0 Mental Math Sayaka subtracts on a number line as follows: She marks the number she is subtracting on a number line. She draws an arrow to show the difference between and the nearest whole She draws an arrow to show the difference between and (= ). She sees that: = + = number ().. Follow Sayaka s steps to find the difference. The first question is started for you. b) 0 0 = 0 =. Find the differences. b) c) = = =. Find the differences mentally: = b) = c) = d) 0 = 0 COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED. Find the difference by following the steps shown. b) = + + =. Find the differences mentally. = b) = c) = d) = Number Sense -0

32 NS- Investigating Fractions and Division Advanced. Write each improper fraction in terms of division. = b) = c) = d) =. Write each mixed number in terms of addition and division. = + b) = c) = d) =. Which improper fraction from question is equivalent to a mixed number from Question? Verify your answer by doing the calculations directly. To add fractions that have a common denominator, we can add the numerators. Example : = Example : 0 + =. Translate the examples in the box into a statement using division. Example : = 0 Example : + =. Translate each statement of addition of fractions into addition of quotients. Verify by dividing and adding. + = + b) + = + 0 c) + + = ( + ) + = = 0 + = INVESTIGATION To add fractions that have a common numerator, can we add the denominators? A. Is + more or less than? How do you know? B. Is + more or less than? How do you know? C. Can + = +? Explain.. The symbol means not equal to. Translate the statements using division instead of fractions. Verify by division that the two sides are not equal COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED Number Sense -

33 NS- Word Problems with Adding and Subtracting Fractions. What fraction of a year is months? =. What fraction of an hour is minutes?. What fraction of a metre is 0 cm?. A cup is 0 ml. What fraction of a cup is 00 ml?. Tania cycled km in the first hour, and kilometres did she cycle in two hours? km in the second hour. How many. Monica ran km in the first hour, km the second hour, and third hour. How many kilometres did she run in three hours? km the. A cafeteria sold cheese pizzas, vegetable pizzas, and at lunchtime. How many pizzas did they sell altogether? deluxe pizzas. Trevor has of an hour to write a test. If he finishes the test in how much time would remain? of an hour then. Tegan bought metres of ribbon. She needs metres to wrap her brother s present and metres to wrap her sister s present. Did she buy enough ribbon? 0. Anne just turned years old. How old was she years ago?. Tabitha ate of a sub and Jordan ate of the sub. How much of the sub was left over? COPYRIGHT 00 JUMP MATH: NOT TO BE COPIED. Sadia picked of a basket of berries, Tashi picked of a basket, Shafma picked of a basket, and Marzuk picked of a basket. Did all their berries fit into baskets?. Soil at a beach consists of sand, clay and salt. If a sample taken is sand and salt, how much of the sample is clay?. Gerome has an hour and a half before he has to leave to play basketball. If it takes him of an hour to do his homework and of an hour to find his uniform, how much time will he have left? Number Sense -

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